Binaural Audio is simply audio recorded by two microphones and has become a blanket term of sorts.
True binaural audio, at least as it was originally defined, is simple a stereo recording made by using two omni microphones and an acoustic shadow. This was invented by the great electrical engineer Alan Blumlein.
These days, binaural audio tends to refer to dummy head recordings or holophonic recordings.
Dummy head recordings are stereo recordings made utilizing a replica of the human head and human ears, with or without the auditory meatus.
A holophonic recording, is a stereo recording made utilizing an object simulating a human head but without using human ears (pinna).
Both dummy head recordings and holophonic recordings reproduce a realistic "you are there" soundstage when played back on headphones. The biggest advantage to the dummy head vs. the holophone is that the dummy head does better with maintaining certain auditory cues more precisely, like front to back and up and down positions.
Both dummy head recordings and holophonic recordings are popular in recording ambiance. This is where they are most used in film, music production and video games. This recorded audio is then mixed in with other recordings to provide the desired effect.
They are rarely used as a sole medium because they allow very little leeway in the area of post-production tinkering. If an artist can't sing or perform properly, these recording techniques will spotlight it and this is a major reason why it's not popular. It's also a minimalist recording technique and these techniques don't always allow producers to use all their associated toys (limiters, pre-amps, EQ, more microphones, etc etc) for better post-production control. Dummy head recordings and holophonic recordings are essentially as good as the recording engineer and artists and the placement of the microphone is critical as is accurate monitoring. Poor choice in position and poor monitoring lead many dummy head recordings and holophonic recordings to be speaker incompatible. Many also feel that the inherent cross-talk in speaker playback produces a flawed reproduction (although I feel it's quite adequate when done right).
How do you know if it's a dummy head recording or holophonic recording? Most clearly state it. Sometimes, you can simply tell by listening or reading the accompanied booklet and seeing what microphone was used.
Hope that helps.